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Cotype for p-Banach spaces. (English) Zbl 0606.46011
Let \(X\) be a real vector space, \(0<p\leq 1\) and a p-convex norm on \(X\), i.e. a function \(\| \cdot \|: X\to R_+\) with the properties:
i) \(\| x\| >0\), for \(x\neq 0\)
ii) \(\| ax\| =| a| \| x\|\), \(a\in {\mathbb{R}}\) and \(x\in X\)
iii) \(\| x+y\|^ p\leq \| x\|^ p+\| y\|^ p\), \(x,y\in X.\)
Then \((X,\| \cdot \|)\) is a p-Banach space. And a p-Banach space \((X,\| \cdot \|)\) is of cotype \(q\), \(0<q\leq +\infty\), if there exists a constant \(C>0:\) \((\sum^{n}_{1}\| x_ i\|^ q)^{1/q}\leq C\int^{1}_{0}\| \sum^{n}_{1}r_ i(t)x_ i\| dt\) for all \(n\in {\mathbb{N}}\) and \(x_ i\in X\), where \(\{r_ i(t)\}^ n_ 1\) are the Rademacher functions on \([0,1]\).
Now in this paper the theorem of Maurey and Pisier for the cotype in p- Banach spaces is proved. Precisely it is proved that for \((X,\| \cdot \|)\) a real p-Banach space and \(q(X)=\inf \{q: X\) is of \(q\)-Rademacher cotype\(\}\ell^{q(X)}\) is finitely representable in \(X\), when \(q(X)\) is real and, for each \(\epsilon >0\), \(c_ 0\) is \((2^{1/p-1}+\epsilon)\) finitely representable in X, when \(q(X)=\infty\).
Reviewer: K.Stathakopoulos
MSC:
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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